Forecasting - time series - smoothing and decomposition methods Smoothing Method as Moving Averages and exponetial methods. The steps for decomposition methods and example of it. Case study for smothing methods in Single Exponential Smoothing, Double Exponential Smoothing and Triple Exponential Smoothing

1. FORECASTING ANALYSIS
TIME SERIES
SMOOTHING AND
DECOMPOSITION
PL 5101
PLANNING ANALYTICAL
METHOD
BY MUH.
AWALUDDIN. T

2. A. BACKGROUND
B. SMOOTHING METHODS
C. DECOMPOSITION METHODS
D. CASE STUDY
1. Introduction
2. Types of Data Pattern
3. Evaluation Model
1. Averaging Methods
2. Exponential Smoothing Methods
1. Explanation
2. Example
1. Smoothing Method by using
Exponential Smoothing
CONTENTS
PL 5101
PLANNING ANALYTICAL
METHOD
FORECASTING ANALYSIS
TIME SERIES
Smoothing and Decomposition Methods

3. A. BACKGROUND
1. INTRODUCTION
 Quantitative forecasting can be applied when three conditions quantitative exist:
1. Information about the past is available.
2. This information can be quantified in the form of numerical data.
3. It can be assumed that some aspects of the past pattern will continue into the future.
2 major types of quantitative forecasting : Time-Series and Regression (Causal) Methods.
The objective of Time – Series is to discover the pattern in the historical data series and
extrapolate the pattern into the future.
 The reasons :
1.The system may not be understood (Makridakis et al, 2008)
2.Explanatory is necessary to be predicted but it is too difficult (Rob J Hyndman, 2009)
3.Only to predict what will happen (Makridakis et al, 2008)
SOURCE:MakridakisEtAl,2008
Look at the data
(Scatter Plot)
Forecast using one or
more techniques
Evaluate the technique
and pick the best one.
Diagram Basics Procedur in Forecasting (source : http://www2.gsu.edu/~dscsss/teaching/mgs3100/sum07/Ch_5.ppt

4. A. BACKGROUND
2. TYPES OF DATA PATTERN
For time series, the most obvious time plot graphical form is a time plot in which the data
are plotted over time.
Four types of time series patterns data patterns can be distinguished: horizontal, seasonal,
cyclical, and trend.
An important step in selecting an appropriate forecasting method is to consider the types
of data patterns, so that the methods most appropriate to those patterns can be utilized.
SOURCE : Spyros Et Al, 1997
The major task of time series analysis is to describe the nature of the past variation of a
variable in order that its future values can be predicted and acted upon accordingly
(Kachigan, 1986)

5. A. BACKGROUND
2. TYPES OF DATA PATTERN
A HORIZONTAL (H) PATTERN
exists when the data values fluctuate around a constant mean or Stationary
SOURCE:MakridakisEtAl,2008
https://www.cengage.com

6. A. BACKGROUND
2. TYPES OF DATA PATTERN
A SEASONAL (S) PATTERN
exists when a series is influenced by seasonal factors or in regular interval.
(e.g., the quarter of the year, the month, or day of the week).
SOURCE:MakridakisEtAl,2008

7. A. BACKGROUND
2. TYPES OF DATA PATTERN
A CYCLICAL (C) PATTERN
exists when the data exhibit rises and falls that are not of a fixed period.
SOURCE:MakridakisEtAl,2008

8. A. BACKGROUND
2. TYPES OF DATA PATTERN
A TREND (T) PATTERN
exists when there is a long-term increase or decrease in the data.
SOURCE:MakridakisEtAl,2008

9. A. BACKGROUND
3. EVALUATION MODEL
SOURCE:MakridaisEtAl,2008
 ME - The arithmetic mean of the errors
 n is the number of forecast errors
 Excel: =AVERAGE(error range)
 Mean Absolute Deviation - MAD
n
Error
n
Forecast)-(Actual
ME
 
n
|Error|
n
Forecast-Actual|
MAD
 
|
 Mean Square Error - MSE
•
• Excel: =SUMSQ(error
range)/COUNT(error range)
Mean Absolute Percentage Error - MAPE
• In general, the lower the error measure (ME,
MAD, MSE) the better the forecasting model
n
(Error)
n
Forecast)-(Actual
MSE
22
 
n
Actual
|Forecast-Actual|
MAPE


%100*

11. B. SMOOTHING METHODS
 The basis of the smoothing methods is the simple weighting or smoothing of
past observations in a time series in order to obtain a forecast for the future.
 If the time series involves a trend (in an upward or downward direction), or a
seasonal effect or both a trend and pattern, we consider a variety of smoothing
methods seasonal effect that seek to improve upon the mean as the forecast for
the next period(s).
 The major advantages of smoothing methods are their low cost, the ease with
which they can be applied, and the speed with which they can be adopted.
 These characteristics make them particularly attractive when a large number
of items are to be forecasted, such as would be the case in many inventory
situations, and when the time horizon is relatively short (less than 1 year).

12. B. SMOOTHING METHODS
STRATEGY FOR APPRAISING ANY OF THE SMOOTHING METHODS OF FORECASTING (SPYROS ET AL, 1997)

13. B. SMOOTHING METHODS
 Pegels (1969) has provide a simple but useful framework for separating trend and seasonal aspects is
whether or not the model should be additive (linear) or multiplicative (non linear) in smoothing methods.

14. B. SMOOTHING METHODS 1. AVERAGING METHODS
a. The Mean & Single Moving Avarage
 The method of the mean is simply
to take the average of all observed
data as the forecast.
THE EQUATION IS :
 The single moving averages method uses
the average of the most recent k data values in
the time series as the forecast for the next
period.
THE EQUATION IS :
It cannot handle trend or seasonality very well, although it can do better than the total
mean. It is useful for time series with a slowly changing mean.

15. B. SMOOTHING METHODS 1. AVERAGING METHODS
c. Double Moving Avarages and Other Moving Avarages
Combination
 The double moving averages method uses
the technique of single average moving with an
adjustment for trend from period t to period t+1
THE EQUATION IS :
 It can handle trend but due to the generally
superior of Exponential Smoothing Methods,
this method not used often for forecasting.
THE EQUATION for Moving Avarages
Method is :
o where m=2k+1. That is, the estimate of the
trend-cycle at time t is obtained by
averaging values of the time series within
k periods of t.
 Latter used in Decomposition Method in
various way for Smoothing not for
Forecasting

16. B. SMOOTHING METHODS 2. EXPONENTIAL SMOOTHING METHODS
 Exponential smoothing method often used to forecast due their simplicity and low cost
It require that certain parameters be defined, and that parameters value lie between 0
and 1
 Exponential smoothing methods provide forecasts using weighted averages of past values of
data and forecast errors.
 In exponential smoothing (as opposed to in moving averages smoothing) older data is given
progressively-less relative weight (importance) whereas newer data is given progressively-
greater weight.
 Four types of Exponential Smoothing :
1. Single Exponential Smoothing (SES)
2.Single Exponential Smoothing –
Adaptive Approach
3. Double Exponential Smoothing
a. Brown One Parameter (Linear method)
b. Holt Two Parameter(Linear method)
4. Triple Exponential Smoothing
a. Brown (Squared method)
b. Holt-Winter (Trend and Seasonality)

17. B. SMOOTHING METHODS 2. EXPONENTIAL SMOOTHING METHODS
 SES method gives the most recent
observation with a weight value (α) and
weighting the most recent previous forecast
with a weight (1-α).
 Used for Horizontal Data Pattern or
Stationary with no Trend
a. Single Exponential Smoothing (SES) and Single Exponential
Smoothing – An Adaptive Approach
 SES – An Adaptive Approach has principle
like SES but α value could change in a
controlled manner, as changes in the
pattern of data occur.
 Used for Horizontal Data Pattern or
Stationary with Trend (Linier)
THE EQUATION and INITIATION :THE EQUATION and INITIATION :

18. B. SMOOTHING METHODS 2. EXPONENTIAL SMOOTHING METHODS
 Double exponential smoothing Brown One
Parameter Linier Method suppose the  is
lies between 0.1 to 0.2 and the  used For
twice smoothing stages.
 Used for Horizontal Data Pattern or
Stationary with Trend (Linier).
b. Double Exponential Smoothing (Brown One Parameter and Holt
Two Paramater
 Double exponential smoothing Holt Two
Parameter Linier Method smooth the
trend seperatelyby using two smoothing
constant (α and γ) between 0 to 1.
 Used for Horizontal Data Pattern or
Stationary with Trend (Linier).
THE EQUATION and INITIATION :THE EQUATION and INITIATION :

19. B. SMOOTHING METHODS 2. EXPONENTIAL SMOOTHING METHODS
c. Triple Exponential Smoothing (Brown One Parameter Quadratic
Method and Winter Three Paramater Trend & Seasonality Method)
 Used for Horizontal Data Pattern or
Stationary with Trend and Seasonality
THE EQUATION and INITIATION :
THE EQUATION and INITIATION :

20. B. SMOOTHING METHODS 2. EXPONENTIAL SMOOTHING METHODS
d. Comparison of Smoothing Method
 If the data series is stationary and
nonseasonal, the Adaptive-Response-Rate
Single Exponential Smoothing is often
preffered as the optimal  will change if the
pattern in basic data is change.
For data series is nonstationary and
nonseasonal, it is preffered to use the
Double Exponetial Smoothing : Brown’s One
Parameter Linier Method due to the easiness
of the method for  value.
If the data series involved a turning points, it
is often recommended to apply the Triple
Exponential Smoothing : Brown’s One-
Parameter Quadratic Method and adjust the
 below 0.1
 Only Triple Exponential Smoothing :
Winter’s Method that widely used for
seasonal data series although it require 3
parameters (,  and ) provided by trial
and error.
 The initialization or the value of the first
forecast (F1) for most of the smoothing
method are the value of first data (X1).

22.  Classical time series decomposition separates a time series into five components: mean, long-
range trend, seasonality, cycle, and randomness.
 The decomposition model is Value = (Mean) x (Trend) x (Seasonality) x (Cycle) x (Random).
 Time series decomposition can be used to separate or decompose a time series into seasonal,
trend, and irregular components.
 A general mathematical model takes the following form:
 In this model, the trend and seasonal and
irregular components are multiplied to give
the value of the time series. Trend is
measured in units of the item being forecast.
However, the seasonal and irregular
components are measured in relative terms,
with values above 1.00 indicating effects
above the trend and values below 1.00
indicating effects below the trend.
Xt = f(It, Tt, Ct, Et),
 Where :
Xt is the Time Series Value (Actual Data) At Period t
It Is the Seasonal component (Index) at period t
Tt is the Trend component (Index) at period t
Ct is the Cycal component (Index) at period t
Et is the Error or Random component at period t
C. DECOMPOSITION METHODS

23. C. DECOMPOSITION METHODS
1. Step 1, For the actual series Xt, compute a moving average whose length,N, is equal to the
length of seasonality;
2. Step 2, Separate the N period moving average (step 1) from the original data series to obtain
trend and cyclicality;
3. Step 3, Isolate the seasonal factors by averaging them for each of the periods making up the
complete length of seasonality;
4. Step 4, Identify the appropriate form of the trend (linear, exponential, etc) and calculate its
value at each period Tt;
5. Step 5, Separate the outcome of step 4 from step 2 to obtain the cyclical factor;
6. Step 6, Separate the seasonality, trend, and cycle from the original data series to isolate the
remaining randomness.
 The process of decomposition methods :
 Commonly, there are two types of decomposition method which are Classical (1920s) Additive
Form that appropriate if the magnitude of the seasonal does not vary and Multicative Form. This
two types have same mathematical model.

24.  To illustrate the process of Multicative Decomposition, we take a data of the television set sales
in 4 year that divided in 4 quarter each year.
C. DECOMPOSITION METHODS
Year Quarter
Sales
(Thousand)
1 1 4.80
1 2 4.10
1 3 6.00
1 4 6.50
2 1 5.80
2 2 5.20
2 3 6.80
2 4 7.40
Continued
3 1 6.00
3 2 5.60
3 3 7.50
3 4 7.80
4 1 6.30
4 2 5.90
4 3 8.00
4 4 8.40
0.00
0.00
0.01
0.01
0.01
0.01
0.01
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Thousands
Television Sale

25. C. DECOMPOSITION METHODS
 Step 1 & 2 : Calculate the Moving Avarages and Substract the moving avarages values to result Seasonality
and Randomness
Year Quarter Sales ( Thousand)
Four-Quarter Moving
Avarage
Centered Moving
Average
1 1 4.80
1 2 4.10 5.350
1 3 6.00 5.600 5.475
1 4 6.50 5.875 5.738
2 1 5.80 6.075 5.975
2 2 5.20 6.300 6.188
2 3 6.80 6.350 6.325
2 4 7.40 6.450 6.400
3 1 6.00 6.625 6.538
3 2 5.60 6.725 6.675
3 3 7.50 6.800 6.763
3 4 7.80 6.875 6.838
4 1 6.30 7.000 6.938
4 2 5.90 7.150 7.075
4 3 8.00
4 4 8.40

26. C. DECOMPOSITION METHODS
 Step 3 : Separates the Randomness from the Seasonality
Quarter Seasonal Irregular
Seasonal
Index
1
0.971 0.918
0.91
0.93
2
0.840 0.839
0.83
0.84
3
1.096 1.075
1.11
1.09
4
1.133 1.156
1.14
1.14
Year Quarter Sales
Four-Quarter
Moving Avarage
Centered
Moving
Average
Deseasonalized
Sales
1 1 4.80 5.15
1 2 4.10 5.350 4.89
1 3 6.00 5.600 5.475 5.49
1 4 6.50 5.875 5.738 5.69
2 1 5.80 6.075 5.975 6.22
2 2 5.20 6.300 6.188 6.21
2 3 6.80 6.350 6.325 6.22
2 4 7.40 6.450 6.400 6.47
3 1 6.00 6.625 6.538 6.44
3 2 5.60 6.725 6.675 6.68
3 3 7.50 6.800 6.763 6.86
3 4 7.80 6.875 6.838 6.82
4 1 6.30 7.000 6.938 6.76
4 2 5.90 7.150 7.075 7.04
4 3 8.00 7.32
4 4 8.40 r 7.35

27. C. DECOMPOSITION METHODS
 Step 4 & 5 : Separates the Trend and the Cycle
 To identify this trend, we will fit a linear
trend equation to the deseasonalized
time series .
The trend line will be fitted to the
deseasonalized data instead of the
original data.
 A linear trend the estimated
regression equation is
 From manual calculation by
excel, the estimated linear trend
equation is

28. C. DECOMPOSITION METHODS
Deseasonalized Sales T - t Y - y (T-t) x (Y-y) (T-t)2
5.15 (7.50) (1.20) 9.006698 56.25
4.89 (6.50) (1.46) 9.46395 42.25
5.49 (5.50) (0.86) 4.742501 30.25
5.69 (4.50) (0.66) 2.991279 20.25
6.22 (3.50) (0.13) 0.448568 12.25
6.21 (2.50) (0.14) 0.357415 6.25
6.22 (1.50) (0.13) 0.195864 2.25
6.47 (0.50) 0.12 -0.06123 0.25
6.44 0.50 0.09 0.043192 0.25
6.68 1.50 0.33 0.501747 2.25
6.86 2.50 0.51 1.274147 6.25
6.82 3.50 0.47 1.65314 12.25
6.76 4.50 0.41 1.836913 20.25
7.04 5.50 0.69 3.80928 30.25
7.32 6.50 0.97 6.285301 42.25
7.35 7.50 1.00 7.478399 56.25
101.60 - 0.00 50.03 340.00
6.35
0.14714
5.10
T 136.000
t 8.50
Y 101.60
y 6.35
 Step 4 & 5 : Separates the Trend and the Cycle

29. C. DECOMPOSITION METHODS
 Step 6 : Isolates the Randomness
 The slope of 0.147 indicates that over the past 16 quarters, the firm averaged a deseasonalized
growth in sales of about 147 sets per quarter. If we assume that the past 16-quarter trend in
sales data is a reasonably good indicator of the future, this equation can be used to develop a
trend projection for future quarters. For example, substituting t 17 into the equation yields
next quarter’s deseasonalized trend projection, T17.
Year Quarter
Deseasonalized Trend
Foorecast
5 1 7.601
2 7.748
3 7.895
4 8.042

30. C. DECOMPOSITION METHODS
 The forecast for 17, 18 , 19 and 20 are
Year Quarter
Deseasonalized Trend
Foorecast
Seasonal
Index
Quarterly
Forecast
(Thousand)
5 1 7.601 0.93 7.086
2 7.748 0.84 6.491
3 7.895 1.09 8.632
4 8.042 1.14 9.195

31. CASE
STUDY

32. The Use of Exponential Smoothing Method to Predict Missing Service E-Report
AHMAD CHUSYAIRI
Information Technology STIKOM PGRI
Banyuwangi
PELSRI RAMADAR N.S.
Information Technology STIKOM PGRI
Banyuwangi
BAGIO
Planning Departement Police Resort
Banyuwangi
In this research examines the selection of an appropriate forecasting model in accordance with
time series data available for predicting the missing reports in a period.
2017 2nd International Conferences on Information Technology, Information Systems and Electrical Engineering (ICITISEE)

33. ALOGARITHM ANALYSIS
 Forecasting missing report at Police Resort
Banyuwangi by using Time Series in
Smoothing Methods.
 The forecasting techniques used are single,
double, and triple exponential smoothing.
 The error value of the predicted data obtained
based on the data being tested using MAD,
MSE, and MAPE method for every single
technique.

34. DATA REPORT

35. VARIABLE USED IN SMOOTHING METHOD
 The value of Alpha, Betha and Gamma
determained by Trial and Error
 The double exponential smoothing method
used Holt’s Two-Parameter Method while
the triple exponential smoothing method
used Winter’s Three-Parameter Trend and
Seasonality Method.
 The actual value used in Single
Exponetial Smoothing was  = 0.6,
not  = 0.8

36. RESULT OF THE RESEARCH
 Conclusion :
The most suitable method in predicting the
moving data up and down (fluctuation)in the
data report is by using the Single Exponential
Smoothing method because it has the error
value of the prediction data using Mean Absolute
Deviation (MAD), and Mean Square Error
(MSE), however triple exponential smoothing
method has a smallest error value using Mean
Absolute Percentage Error (MAPE).
o Review :
1. SES actualy was the lowest one only in
MAD not in MSE (DES was the lower).
2. I try using other method by using DES:
Brown’s Linier Method and TES Brown
Quadratic Method which is easier to
apply than DES Holt’s Method or TES
Winter’s Method. The DES Brown’s
Method is as simple as SES but provide
lower result in MAD and MSE
DES :  = 0.2 and Initialization S’=S’’= X
TES :  = 0.15 and Initialization S’=S’’=S’’’=X
No. Method MAD MSE MAPE
1 DES Brown 4.92 725.89 282.29
2 TES Brown 4.17 521.68 295.58

37. REFFERENCE
 Spyros G. Makridakis, Steven C. Wheelwright, Rob J Hyndman - Forecasting
Methods and Applications - Wiley (1997)
 The Use of Exponential Smoothing Method to Predict Missing Service E-
Report - Ahmad Chusyairi, Pelsri Ramadar N.S. and Bagio,
https://ieeexplore.ieee.org/
 http://www2.gsu.edu/~dscsss/teaching/mgs3100/sum07/Ch_5.ppt
 https://otexts.org/fpp2/
 https://ec.europa.eu/eurostat/statistics-explained
 http://www.businessdictionary.com/definition/exponential-smoothing
 http://www.ncss.com
 https://www.cengage.com/resource_uploads/downloads/0840062389_34
7257.pdf
 https://arumprimandari.files.wordpress.com/2015/03/course-5_pegels-
classification.pdf